Satellite Geodesy
2nd completely revised and extended edition
≥
Günter Seeber,Univ. Prof. Dr.-Ing. Institut für Erdmessung Universität Hannover Schneiderberg 50 30167 Hannover Germany 1st edition 1993
This book contains 281 figures and 64 tables.
앪앝 Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data
Seeber,Günter,1941⫺
[Satellitengeodäsie. English]
Satellite geodesy : foundations,methods,and applications / Gün-ter Seeber.⫺ 2nd completely rev. and extended ed.
p. cm.
Includes bibliographical references and index. ISBN 3-11-017549-5 (alk. paper)
1. Satellite geodesy. I. Title. QB343 .S4313 2003
526⬘.1⫺dc21
2003053126
ISBN 3-11-017549-5
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at⬍http://dnb.ddb.de⬎. 쑔 Copyright 2003 by Walter de Gruyter GmbH & Co. KG,10785 Berlin
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Printed in Germany
Cover design: Rudolf Hübler,Berlin
Typeset using the authors TEX files: I. Zimmermann,Freiburg Printing and binding: Hubert & Co. GmbH & Co. Kg,Göttingen
Methods of satellite geodesy are increasingly used in geodesy, surveying engineering, and related disciplines. In particular, the modern development of precise and opera-tional satellite based positioning and navigation techniques have entered all fields of geosciences and engineering. A growing demand is also evident for fine-structured gravity field models from new and forthcoming satellite missions and for the monitor-ing of Earth’s rotation in space. For many years I have had the feelmonitor-ing that there is a definite need for a systematic textbook covering the whole subject, including both its foundations and its applications. It is my intention that this book should, at least in part, help to fulfill this requirement.
The material presented here is partly based on courses taught at the University of Hannover since 1973 and on guest lectures given abroad. It is my hope that this mate-rial can be used at other universities for similar courses. This book is intended to serve as a text for advanced undergraduates and for graduates, mainly in geodesy, survey-ing engineersurvey-ing, photogrammetry, cartography and geomatics. It is also intended as a source of information for professionals who have an interest in the methods and results of satellite geodesy and who need to acquaint themselves with new developments. In addition, this book is aimed at students, teachers, professionals and scientists from related fields of engineering and geosciences, such as terrestrial and space navigation, hydrography, civil engineering, traffic control, GIS technology, geography, geology, geophysics and oceanography. In line with this objective, the character of the book falls somewhere between that of a textbook and that of a handbook. The background required is an undergraduate level of mathematics and elementary mathematical statis-tics. Because of rapid and continuous developments in this field, it has been necessary to be selective, and to give greater weight to some topics than to others. Particular importance has been attached to the fundamentals and to the applications, especially to the use of artificial satellites for the determination of precise positions. A compre-hensive list of references has been added for further reading to facilitate deeper and advanced studies.
The first edition of this book was published in 1993 as an English translation and update of the book “Satellitengeodäsie”, that was printed in the German language in 1989. The present edition has been completely revised and significantly extended. The fundamental structure of the first edition has been maintained to facilitate continuity of teaching; however, outdated material has been removed and new material has been included. All chapters have been updated and some have been re-written. The overall status is autumn 2002 but some of the most recent technological developments to March 2003 have been included.
Extensions and updates mainly pertain to reference coordinate systems and refer-ence frames [2.2], signal propagation [2.3], directions with CCD technology [5.2], the Global Positioning System (GPS) and GNSS [7], satellite laser ranging [8], satellite
altimetry [9], gravity field missions [10] and applications [12]. In particular, the chap-ter on GPS and GNSS [7] has been almost completely re-written and now covers about 200 pages. Together with chapters [2], [3], and [12], it forms a comprehensive GPS manual on its own. New technological developments of the space and user segment are included, as is the current state of data analysis and error budget. Differential GPS and permanent reference networks are now treated in a comprehensive section of their own [7.5]. GLONASS and the forthcoming GALILEO are included in a new section on GNSS [7.7].
Gravity field missions like CHAMP, GRACE and GOCE, because of their increas-ing importance, are dealt with in a new chapter [10]. VLBI, together with the new inclusion of interferometric SAR, form another new chapter [11]. Coverage of histor-ical techniques like photographic camera observations [5] and Transit Doppler [6] has been considerably reduced. The basic principles, however, are still included because of their historical importance and because they are shared by new technologies like CCD cameras [5.2] and DORIS [6.7]. The geodetic history of Transit Doppler tech-niques, in addition, is an excellent source for understanding the evolution and basic concepts of the GPS. The chapter on applications, now renumbered [12], has been updated to include modern developments and a new section on the combination of geodetic space techniques [12.5]. International services of interest to satellite geodesy have been included, namely the IGS [7.8.1], the ILRS [8.5.1], the IVS [11.1.3], and the IERS [12.4].
The bibliography has been updated and expanded considerably by adding an in-creased number of English language references. The total number of references is now reaching 760, about half of which are new in this edition.
Many of the examples within this book are based on field projects and research work carried out in collaboration with my graduate students, doctorate candidates and scientific colleagues at the University of Hannover over more than 20 years. I would like to thank all these individuals for their long standing cooperation and the many fruitful discussions I have had with them. In addition, the help of the staff at the Institut für Erdmessung is gratefully acknowledged. Most figures have been redrawn by cand. geod. Anke Daubner and Dipl.-Ing. Wolfgang Paech.
My sincere thanks for checking and correcting the English language go to Dr. Graeme Eagles of the Alfred Wegener Institut für Polar- und Meeresforschung, Bremerhaven. I should also like to thank the many colleagues from all over the world who helped to improve the book through their comments on the first edition, and the individuals and organizations who provided illustrations.
Finally my gratitude goes to my wife Gisela for her never ending support and under-standing. The publisher remained excellently cooperative throughout the preparation of this book. My cordial thanks go to Dr. Manfred Karbe, Dr. Irene Zimmermann, and the staff at Walter de Gruyter.
Preface vii
Abbreviations xvii
1 Introduction 1
1.1 Subject of Satellite Geodesy . . . 1
1.2 Classification and Basic Concepts of Satellite Geodesy . . . 3
1.3 Historical Development of Satellite Geodesy . . . 5
1.4 Applications of Satellite Geodesy . . . 7
1.5 Structure and Objective of the Book . . . 9
2 Fundamentals 10 2.1 Reference Coordinate Systems . . . 10
2.1.1 Cartesian Coordinate Systems and Coordinate Transformations 10 2.1.2 Reference Coordinate Systems and Frames in Satellite Geodesy 12 2.1.2.1 Conventional Inertial Systems and Frames . . . 13
2.1.2.2 Conventional Terrestrial Systems and Frames . . . . 15
2.1.2.3 Relationship between CIS and CTS . . . 17
2.1.3 Reference Coordinate Systems in the Gravity Field of Earth . 21 2.1.4 Ellipsoidal Reference Coordinate Systems . . . 23
2.1.5 Ellipsoid, Geoid and Geodetic Datum . . . 25
2.1.6 World Geodetic System 1984 (WGS 84) . . . 28
2.1.7 Three-dimensional Eccentricity Computation . . . 30
2.2 Time . . . 31
2.2.1 Basic Considerations . . . 31
2.2.2 Sidereal Time and Universal Time . . . 32
2.2.3 Atomic Time . . . 35
2.2.4 Ephemeris Time, Dynamical Time, Terrestrial Time . . . 37
2.2.5 Clocks and Frequency Standards . . . 39
2.3 Signal Propagation . . . 42
2.3.1 Some Fundamentals of Wave Propagation . . . 43
2.3.1.1 Basic Relations and Definitions . . . 43
2.3.1.2 Dispersion, Phase Velocity and Group Velocity . . . 45
2.3.1.3 Frequency Domains . . . 46
2.3.2 Structure and Subdivision of the Atmosphere . . . 48
2.3.3 Signal Propagation through the Ionosphere and the Troposphere 52 2.3.3.1 Ionospheric Refraction . . . 54
3 Satellite Orbital Motion 62
3.1 Fundamentals of Celestial Mechanics, Two-Body Problem . . . 62
3.1.1 Keplerian Motion . . . 63
3.1.2 Newtonian Mechanics, Two-Body Problem . . . 66
3.1.2.1 Equation of Motion . . . 66
3.1.2.2 Elementary Integration . . . 69
3.1.2.3 Vectorial Integration . . . 74
3.1.3 Orbit Geometry and Orbital Motion . . . 77
3.2 Perturbed Satellite Motion . . . 82
3.2.1 Representation of the Perturbed Orbital Motion . . . 84
3.2.1.1 Osculating and Mean Orbital Elements . . . 84
3.2.1.2 Lagrange’s Perturbation Equations . . . 85
3.2.1.3 Gaussian Form of Perturbation Equation . . . 87
3.2.2 Disturbed Motion due to Earth’s Anomalous Gravity Field . . 88
3.2.2.1 Perturbation Equation and Geopotential . . . 89
3.2.2.2 Perturbations of the Elements . . . 94
3.2.2.3 Perturbations Caused by the Zonal CoefficientsJn . 96 3.2.3 Other Perturbations . . . 98
3.2.3.1 Perturbing Forces Caused by the Sun and Moon . . 98
3.2.3.2 Solid Earth Tides and Ocean Tides . . . 101
3.2.3.3 Atmospheric Drag . . . 102
3.2.3.4 Direct and Indirect Solar Radiation Pressure . . . . 104
3.2.3.5 Further Perturbations . . . 105
3.2.3.6 Resonances . . . 107
3.2.4 Implications of Perturbations on Selected Satellite Orbits . . . 108
3.3 Orbit Determination . . . 109
3.3.1 Integration of the Undisturbed Orbit . . . 110
3.3.2 Integration of the Perturbed Orbit . . . 114
3.3.2.1 Analytical Methods of Orbit Integration . . . 114
3.3.2.2 Numerical Methods of Orbit Integration . . . 116
3.3.2.3 Precise Orbit Determination with Spaceborne GPS . 119 3.3.3 Orbit Representation . . . 120
3.3.3.1 Ephemeris Representation for Navigation Satellites 121 3.3.3.2 Polynomial Approximation . . . 122
3.3.3.3 Simplified Short Arc Representation . . . 124
3.4 Satellite Orbits and Constellations . . . 126
3.4.1 Basic Aspects . . . 126
3.4.2 Orbits and Constellations . . . 128
4 Basic Observation Concepts and Satellites Used in Geodesy 135
4.1 Satellite Geodesy as a Parameter Estimation Problem . . . 135
4.2 Observables and Basic Concepts . . . 139
4.2.1 Determination of Directions . . . 139
4.2.2 Determination of Ranges . . . 141
4.2.3 Determination of Range Differences (Doppler method) . . . . 143
4.2.4 Satellite Altimetry . . . 144
4.2.5 Determination of Ranges and Range-Rates (Satellite-to-Satellite Tracking) . . . 144
4.2.6 Interferometric Measurements . . . 145
4.2.7 Further Observation Techniques . . . 147
4.3 Satellites Used in Geodesy . . . 147
4.3.1 Basic Considerations . . . 147
4.3.2 Some Selected Satellites . . . 149
4.3.3 Satellite Subsystems . . . 152
4.3.3.1 Drag Free Systems . . . 152
4.3.3.2 Attitude Control . . . 153
4.3.3.3 Navigation Payload, PRARE . . . 154
4.3.4 Planned Satellites and Missions . . . 156
4.4 Some Early Observation Techniques (Classical Methods) . . . 158
4.4.1 Electronic Ranging SECOR . . . 159
4.4.2 Other Early Observation Techniques . . . 160
5 Optical Methods for the Determination of Directions 161 5.1 Photographic Determination of Directions . . . 161
5.1.1 Satellites used for Camera Observations . . . 162
5.1.2 Satellite Cameras . . . 163
5.1.3 Observation and Plate Reduction . . . 164
5.1.4 Spatial Triangulation . . . 169
5.1.5 Results . . . 170
5.2 Directions with CCD Technology . . . 172
5.2.1 Image Coordinates from CCD Observations . . . 172
5.2.2 Star Catalogs, Star Identification and Plate Reduction . . . 174
5.2.3 Applications, Results and Prospects . . . 176
5.3 Directions from Space Platforms . . . 176
5.3.1 Star Tracker . . . 177
5.3.2 Astrometric Satellites, HIPPARCOS . . . 177
5.3.3 Planned Missions . . . 178
6 Doppler Techniques 181 6.1 Doppler Effect and Basic Positioning Concept . . . 183
6.2 One Successful Example: The Navy Navigation Satellite System . . 186
6.2.1 System Architecture . . . 187
6.3 Doppler Receivers . . . 190
6.3.1 Basic concept . . . 190
6.3.2 Examples of Doppler Survey Sets . . . 192
6.4 Error Budget and Corrections . . . 193
6.4.1 Satellite Orbits . . . 194
6.4.2 Ionospheric and Tropospheric Refraction . . . 195
6.4.3 Receiver System . . . 196
6.4.4 Earth Rotation and Relativistic Effects . . . 197
6.4.5 Motion of the Receiver Antenna . . . 198
6.5 Observation Strategies and Adjustment Models . . . 199
6.5.1 Extended Observation Equation . . . 199
6.5.2 Single Station Positioning . . . 201
6.5.3 Multi-Station Positioning . . . 202
6.6 Applications . . . 203
6.6.1 Applications for Geodetic Control . . . 204
6.6.2 Further Applications . . . 205
6.7 DORIS . . . 207
7 The Global Positioning System (GPS) 211 7.1 Fundamentals . . . 211
7.1.1 Introduction . . . 211
7.1.2 Space Segment . . . 213
7.1.3 Control Segment . . . 217
7.1.4 Observation Principle and Signal Structure . . . 218
7.1.5 Orbit Determination and Orbit Representation . . . 222
7.1.5.1 Determination of the Broadcast Ephemerides . . . . 222
7.1.5.2 Orbit Representation . . . 223
7.1.5.3 Computation of Satellite Time and Satellite Coordinates . . . 225
7.1.5.4 Structure of the GPS Navigation Data . . . 227
7.1.6 Intentional Limitation of the System Accuracy . . . 229
7.1.7 System Development . . . 230
7.2 GPS Receivers (User Segment) . . . 234
7.2.1 Receiver Concepts and Main Receiver Components . . . 234
7.2.2 Code Dependent Signal Processing . . . 239
7.2.3 Codeless and Semicodeless Signal Processing . . . 240
7.2.4 Examples of GPS receivers . . . 243
7.2.4.1 Classical Receivers . . . 243
7.2.4.2 Examples of Currently Available Geodetic Receivers . . . 245
7.2.4.3 Navigation and Handheld Receivers . . . 248
7.2.5 Future Developments and Trends . . . 250
7.3.1 Observables . . . 252
7.3.1.1 Classical View . . . 252
7.3.1.2 Code and Carrier Phases . . . 255
7.3.2 Parameter Estimation . . . 258
7.3.2.1 Linear Combinations and Derived Observables . . . 258
7.3.2.2 Concepts of Parametrization . . . 265
7.3.2.3 Resolution of Ambiguities . . . 269
7.3.3 Data Handling . . . 277
7.3.3.1 Cycle Slips . . . 277
7.3.3.2 The Receiver Independent Data Format RINEX . . 281
7.3.4 Adjustment Strategies and Software Concepts . . . 283
7.3.5 Concepts of Rapid Methods with GPS . . . 289
7.3.5.1 Basic Considerations . . . 289
7.3.5.2 Rapid Static Methods . . . 290
7.3.5.3 Semi Kinematic Methods . . . 292
7.3.5.4 Pure Kinematic Method . . . 294
7.3.6 Navigation with GPS . . . 295
7.4 Error Budget and Corrections . . . 297
7.4.1 Basic Considerations . . . 297
7.4.2 Satellite Geometry and Accuracy Measures . . . 300
7.4.3 Orbits and Clocks . . . 304
7.4.3.1 Broadcast Ephemerides and Clocks . . . 304
7.4.3.2 Precise Ephemerides and Clocks, IGS . . . 307
7.4.4 Signal Propagation . . . 309
7.4.4.1 Ionospheric Effects on GPS Signals . . . 309
7.4.4.2 Tropospheric Propagation Effects . . . 314
7.4.4.3 Multipath . . . 316
7.4.4.4 Further Propagation Effects, Diffraction and Signal Interference . . . 319
7.4.5 Receiving System . . . 320
7.4.5.1 Antenna Phase Center Variation . . . 320
7.4.5.2 Other Error Sources Related to the Receiving System . . . 323
7.4.6 Further Influences, Summary, the Issue of Integrity . . . 323
7.5 Differential GPS and Permanent Reference Networks . . . 325
7.5.1 Differential GPS (DGPS) . . . 326
7.5.1.1 DGPS Concepts . . . 326
7.5.1.2 Data Formats and Data Transmission . . . 329
7.5.1.3 Examples of Services . . . 332
7.5.2 Real Time Kinematic GPS . . . 336
7.5.3 Multiple Reference Stations . . . 338
7.5.3.1 Wide Area Differential GPS . . . 339
7.6 Applications . . . 345
7.6.1 Planning and Realization of GPS Observation . . . 345
7.6.1.1 Setting Up an Observation Plan . . . 346
7.6.1.2 Practical Aspects in Field Observations . . . 348
7.6.1.3 Observation Strategies and Network Design . . . . 350
7.6.2 Possible Applications and Examples of GPS Observations . . 356
7.6.2.1 Geodetic Control Surveys . . . 357
7.6.2.2 Geodynamics . . . 362
7.6.2.3 Height Determination . . . 366
7.6.2.4 Cadastral Surveying, Geographic Information Systems . . . 368
7.6.2.5 Fleet Management, Telematics, Location Based Services . . . 371
7.6.2.6 Engineering and Monitoring . . . 372
7.6.2.7 Precise Marine Navigation, Marine Geodesy, and Hydrography . . . 375
7.6.2.8 Photogrammetry, Remote Sensing, Airborne GPS . 378 7.6.2.9 Special Applications of GPS . . . 380
7.7 GNSS – Global Navigation Satellite System . . . 383
7.7.1 GLONASS . . . 384
7.7.2 GPS/GLONASS Augmentations . . . 392
7.7.3 GALILEO . . . 393
7.8 Services and Organizations Related to GPS . . . 397
7.8.1 The International GPS Service (IGS) . . . 397
7.8.2 Other Services . . . 401
8 Laser Ranging 404 8.1 Introduction . . . 404
8.2 Satellites Equipped with Laser Reflectors . . . 406
8.3 Laser Ranging Systems and Components . . . 411
8.3.1 Laser Oscillators . . . 411
8.3.2 Other System Components . . . 412
8.3.3 Currently Available Fixed and Transportable Laser Systems . 414 8.3.4 Trends in SLR System Developments . . . 416
8.4 Corrections, Data Processing and Accuracy . . . 418
8.4.1 Extended Ranging Equation . . . 418
8.4.2 Data Control, Data Compression, and Normal Points . . . 422
8.5 Applications of Satellite Laser Ranging . . . 424
8.5.1 Realization of Observation Programs, International Laser Ranging Service (ILRS) . . . 424
8.5.2 Parameter Estimation . . . 427
8.5.3 Earth Gravity Field, Precise Orbit Determination (POD) . . . 428
8.5.5 Earth Rotation, Polar Motion . . . 432
8.5.6 Other applications . . . 435
8.6 Lunar Laser Ranging . . . 436
8.7 Spaceborne Laser . . . 441
9 Satellite Altimetry 443 9.1 Basic Concept . . . 443
9.2 Satellites and Missions . . . 444
9.3 Measurements, Corrections, Accuracy . . . 451
9.3.1 Geometry of Altimeter Observations . . . 451
9.3.2 Data Generation . . . 452
9.3.3 Corrections and Error Budget . . . 454
9.4 Determination of the Mean Sea Surface . . . 460
9.5 Applications of Satellite Altimetry . . . 461
9.5.1 Geoid and Gravity Field Determination . . . 462
9.5.2 Geophysical Interpretation . . . 464
9.5.3 Oceanography and Glaciology . . . 465
10 Gravity Field Missions 469 10.1 Basic Considerations . . . 469
10.2 Satellite-to-Satellite Tracking (SST) . . . 473
10.2.1 Concepts . . . 473
10.2.2 High-Low Mode, CHAMP . . . 476
10.2.3 Low-Low Mode, GRACE . . . 477
10.3 Satellite Gravity Gradiometry . . . 480
10.3.1 Concepts . . . 480
10.3.2 GOCE mission . . . 482
11 Related Space Techniques 485 11.1 Very Long Baseline Interferometry . . . 485
11.1.1 Basic Concept, Observation Equations, and Error Budget . . . 485
11.1.2 Applications . . . 491
11.1.3 International Cooperation, International VLBI Service (IVS) . 496 11.1.4 VLBI with Satellites . . . 498
11.2 Interferometric Synthetic Aperture Radar (InSAR) . . . 500
11.2.1 Basic Concepts, Synthetic Aperture Radar (SAR) . . . 500
11.2.2 Interferometric SAR . . . 502
11.2.3 Differential Radar Interferometry . . . 505
12 Overview and Applications 506 12.1 Positioning . . . 506
12.1.1 Concepts, Absolute and Relative Positioning . . . 506
12.1.2 Global and Regional Networks . . . 510
12.2 Gravity Field and Earth Models . . . 514
12.2.1 Fundamentals . . . 514
12.2.2 Earth Models . . . 519
12.3 Navigation and Marine Geodesy . . . 523
12.3.1 Possible Applications and Accuracy Requirements in Marine Positioning . . . 523
12.3.2 Marine Positioning Techniques . . . 524
12.4 Geodynamics . . . 527
12.4.1 Recent Crustal Movements . . . 527
12.4.2 Earth Rotation, Reference Frames, IERS . . . 529
12.5 Combination of Geodetic Space Techniques . . . 534
12.5.1 Basic Considerations . . . 534
12.5.2 Fundamental Stations . . . 535
12.5.3 Integrated Global Geodetic Observing System (IGGOS) . . . 537
References 539
ACP Area Correction Parameter ADOS African Doppler Survey
AI I Accuracy Improvement Initiative APL Applied Physics Laboratory ARP Antenna Reference Point
AS Anti Spoofing
ASIC Application-Specific Integrated Circuit
BCRS Barycentric Celestial Reference System
BIH Bureau International de l’Heure BIPM Bureau International des poids et
Mésures
BKG Bundesamt für Kartographie und Geodäsie
BPS Bits Per Second
BPSK Binary Phase Shift Keying CACS Canadian Active Control System CAD Computer Assisted Design CBIS Central Bureau (IGS) Information
System
CCD Charge Coupled Device CDP Crustal Dynamics Program CEP Celestial Ephemeris Pole CEP Circular Error Probable
CIO Conventional International Origin CIP Celestial Intermediate Pole CIS Conventional Inertial (Reference)
System
CNES Centre National d’Études Spatiales
CONUS Continental U.S. CORS Continuously Operating
Reference Station CPU Central Processing Unit CRF Celestial Reference Frame CRS Celestial Reference System
CTP Conventional Terrestrial Pole CTS Conventional Terrestrial
(Refer-ence) System
DÖDOC German Austrian Doppler Campaign
DD Double Difference DEM Digital Elevation Model DGFI Deutsches Geodätisches
Forschungsinstitut DGPS Differential GPS DOD Department of Defence DOP Dilution of Precision DOY Day Of the Year
DRMS Distance Root Mean Square EDOC European Doppler Campaign EGM96 Earth Gravitational Model 1996 EGNOS European Geostationary
Naviga-tion Overlay System EOP Earth Orientation Parameter EOS Earth Observing System EPS Real-Time Positioning Service
(SAPOS)
ERM Exact Repeat Mission ERP Earth Rotation Parameter ESA European Space Agency ESNP European Satellite Navigation
Program
EU European Union
FAA Federal Aviation Administration FAGS Federation of Astronomical and
Geophysical Data Analysis Services
FIG Fédération Internationale des Géomètres
FK5 Fifth Fundamental Catalogue FOC Full Operational Capability FRNP Federal Radio Navigation Plan
GAST Greenwich Apparent Sidereal Time
GCRS Geocentric Celestial Reference System
GDR Geophysical Data Record GEM Goddard Earth Model GEO Geostationary Orbit
GFO GEOSAT Follow On
GFZ GeoForschungsZentrum Potsdam GIC GPS Integrity Channel
GIS Geo Information System GLAS Geoscience Laser Altimeter
System
GM Geodetic Mission
GMST Greenwich Mean Sidereal Time GNSS Global Navigation Satellite
System
GRGS Groupe de Recherche de Géodésie Spatiale
GRS80 Geodetic Reference System 1980 GSFC Goddard Space Flight Center HEPS High Precision Real-Time
Posi-tioning Service (SAPOS) IAU International Astronomical Union ICD Interface Control Document ICO Intermediate Circular Orbit ICRF International Celestial Reference
Frame
ICRS International Celestial Reference System
IDS International DORIS Service IERS International Earth Rotation and
Reference Systems Service IF Intermediate Frequency
IGEB Interagency GPS Executive Board IGN Institut Géographique National IGS International GPS Service IGSO Inclined Geo-synchronous Orbit ILRS International Laser Ranging
Service
ILS International Latitude Service INSAR Interferometric SAR
ION Institute of Navigation IPMS International Polar Motion
Service
IRIS International Radio Interferomet-ric Surveying
IRM IERS Reference Meridian IRP IERS Reference Pole IRV Inter-Range Vector
ITRF International Terrestrial Reference Frame
IUGG International Union of Geodesy and Geophysics
IVS International VLBI Service
JD Julian Date
JGM Joint Gravity Model
JGR Journal of Geophysical Research JPL Jet Propulsion Laboratory LADGPS Local Area Differential GPS LAN Longitude of Ascending Node LBS Location Based Service LEO Low Earth Orbiter LLR Lunar Laser Ranging LOD Length of Day MAS Milli Arc Second MEO Medium Earth Orbit
MERIT Monitoring Earth Rotation and Intercomparison of Techniques MJD Modified Julian Date
MSAS Multifunctional Satellite-based Augmentation System NAD North American Datum NASA National Aeronautics and Space
Administration
Nd:YAG Neodymium Yttrium Aluminium Garnet
NDGPS Nationwide Differential Global Positioning System
NEOS National Earth Orientation Service
NGS National Geodetic Survey NIMA National Imagery and Mapping
NIST National Institute of Standards NOAA National Oceanic and
Atmo-spheric Administration OCS Operational Control Segment PCV Phase Center Variation PDA Personal Digital Assistant PDGPS Precise Differential GPS PDOP Position Dilution of Precision PE Precise Ephemerides POD Precise Orbit Determination PPP Precise Point Positioning PPS Precise Positioning Service PRN Pseudo Random Noise PTB Physikalisch Technische
Bundesanstalt RA Radar Altimeter RDS Radio Data System RF Radio Frequency RMS Root Mean Square Error RNAAC Regional Network Associate
Analysis Center
RTCM Radio Technical Commission for Marine Sciences
RTK Real Time Kinematic SA Selective Availability SAD South American Datum SAO Smithsonian Astrophysical
Observatory
SAPOS Satellite Positioning Service SAR Search And Rescue SAR Synthetic Aperture Radar SBAS Satellite Based Augmentation
System
SEP Spherical Error Probable SGG Satellite Gravity Gradiometry SI International System of Units SIR Shuttle Imaging Radar SIS Signal in Space
SISRE Signal in Space Range Error SLR Satellite Laser Ranging
SNR Signal-to-Noise Ratio
SPAD Single Photon Avalanche Diode SPS Standard Positioning Service SST Satellite-to-Satellite Tracking SST Sea Surface Topography SV Space Vehicle
SVN Space Vehicle Number SWH Significant Wave Height
T/P TOPEX/POSEIDON
TAI International Atomic Time TCB Barycentric Coordinate Time TCG Geocentric Coordinate Time TDB Barycentric Dynamical Time TDT Terrestrial Dynamical Time TEC Total Electron Content TECU Total Electron Content Unit TID Travelling Ionospheric
Disturbances
TIGO Transportable Integrated Geodetic Observatory
TRF Terrestrial Reference Frame TT Terrestrial Time
TTFA Time To Fix Ambiguities UEE User Equipment Error UERE User Equivalent Range Error URE User Range Error
USCG U.S. Coast Guard USNO U.S. Naval Observatory UT Universal Time
UTC Universal Time Coordinated VLBA Very Long Baseline Array VLBI Very Long Baseline
Interferometry
VRS Virtual Reference Station VSOP VLBI Space Observatory
Program
VTEC Vertical Electron Content WAAS Wide Area Augmentation System WADGPS Wide Area Differential GPS
1.1
Subject of Satellite Geodesy
Following the classical definition of Helmert (1880/1884), geodesy is the science of the measurement and mapping of the Earth’s surface. This definition includes the determination of the terrestrial external gravity field, as well as the surface of the ocean floor, cf. (Torge, 2001). Satellite Geodesy comprises the observational and computational techniques which allow the solution of geodetic problems by the use of precise measurements to, from, or between artificial, mostly near-Earth, satellites. Further to Helmert’s definition, which is basically still valid, the objectives of satellite geodesy are today mainly considered in a functional way. They also include, because of the increasing observational accuracy, time-dependent variations.
The basic problems are
1. determination of precise global, regional and local three-dimensional positions (e.g. the establishment of geodetic control)
2. determination of Earth’s gravity field and linear functions of this field (e.g. a precise geoid)
3. measurement and modeling of geodynamical phenomena (e.g. polar motion, Earth rotation, crustal deformation).
The use of artificial satellites in geodesy has some prerequisites; these are basically a comprehensive knowledge of the satellite motion under the influence of all acting forces as well as the description of the positions of satellites and ground stations in suit-able reference frames. Consequently satellite geodesy belongs to the domain of basic sciences. On the other hand, when satellite observations are used for solving various problems satellite geodesy can be assigned to the field of applied sciences. Consider-ing the nature of the problems, satellite geodesy belongs equally to geosciences and to engineering sciences.
By virtue of their increasing accuracy and speed, the methods and results of satellite geodesy are used more and more in other disciplines like e.g. geophysics, oceanography and navigation, and they form an integral part of geoinformatics.
Since the launch of the first artificial satellite, SPUTNIK-1, on October 4, 1957, satellite geodesy has developed into a self-contained field in geodetic teaching and research, with close relations and interactions with adjacent fields (Fig. 1.1). The assignments and contents are due to historical development.
In Geodetic Astronomy, based on the rules of Spherical Astronomy, the orientation of the local gravity vector (geographical longitude, geographical latitude ), and the astronomical azimuthA of a terrestrial mark are determined from the observation of natural celestial bodies, particularly fixed stars. By Gravimetry we mean the measure-ment of gravity (gravity intensityg) which is the magnitude of the gravity acceleration vectorg (Torge, 1989). With Terrestrial Geodetic Measurements horizontal angles,
Figure 1.1. Main relations between geodetic fields of teaching and research
distances, zenith angles, and levelled height differences are provided, and serve for the determination of surface point locations. Satellite Geodesy, finally, is based on the observation of artificial celestial bodies. Directions, ranges, and range-rates are determined between Earth surface locations and satellites or between satellites. Some measurements, for instance accelerations, are taken within the satellites themselves.
The results of geodetic-astronomic or gravimetric observations are used within the field of Astronomical and Physical Geodesy for the determination of the figure and gravity field of Earth (Torge, 2001). In German, this classical domain is called Erdmessung (Torge, 2003) and corresponds to the concept of Global Geodesy in the English language. The main problems are the determination of a mean Earth ellipsoid and a precise geoid (cf. [2.1.5]).
The determination of coordinates in ellipsoidal or three-dimensional coordinate systems, mainly based on terrestrial geodetic measurements, is treated within the field of Mathematical Geodesy. Alternate expressions for this domain are Geometrical Geodesy or, in German, Landesvermessung, e.g. Großmann (1976). The separate classification of observation- and computation techniques, as developed within the classical fields of geodetic teaching and practice, has not occured to the same extent in satellite geodesy. Here, observation, computation, and analysis are usually treated to-gether. As far as global problems are concerned, satellite geodesy contributes to global geodesy, for example to the establishment of a global reference frame. In regional and local problems, satellite geodesy forms part of surveying and geoinformatics.
Conversely, the fields of mathematical geodesy and geodetic astronomy provide important foundations in satellite geodesy with respect to reference systems. The same is true for the field of astronomical and physical geodesy, which provides infor-mation on Earth’s gravity field. Due to these close interactions, a sharp separation of the different fields in geodesy becomes more and more difficult, and it is no longer significant.
A combined consideration of all geodetic observables in a unified concept was developed rather early within the field of Integrated Geodesy, e.g. Hein (1983). It
finds a modern realization in the establishment of integrated geodetic-geodynamic observatories (see [12.5], Rummel et al. (2000))
The term Satellite Geodesy is more restrictive than the French denomination Géodésie Spatiale or the more general expression Geodetic Space Techniques. The latter term includes the geodetic observation of the Moon, as well as the use of planets and objects outside the solar system, for instance in radio interferometry. Occasionally the term Global Geodesy is used, where global stands for both global measurement techniques and for global applications.
In this book the term Satellite Geodesy is employed, because it is in common usage, and because artificial near-Earth satellites are almost exclusively utilized for the observations which are of interest in applied geodesy. Where necessary, other space techniques are dealt with.
1.2
Classification and Basic Concepts of Satellite Geodesy
The importance of artificial satellites in geodesy becomes evident from the following basic considerations.
(1) Satellites can be used as high orbiting targets, which are visible over large distances. Following the classical concepts of Earth-bound trigonometric networks, the satellites may be regarded as “fixed” control points within large-scale or global three-dimensional networks. If the satellites are observed simultaneously from different
New Station
P1 P2
P3 N
Figure 1.2. Geometrical method; the satellite is a high target
ground stations, it is of no importance that the orbits of artificial satellites are governed by gravitational forces. Only the property that they are targets at high altitudes is used. This purely geometric consideration leads to the geometrical method of satellite geodesy. The con-cept is illustrated in Fig. 1.2. It has been realized in its purest form through the BC4 World Network (see [5.1.5]).
Compared with classical techniques, the main advantage of the satellite meth-ods is that they can bridge large dis-tances, and thus establish geodetic ties between continents and islands. All ground stations belonging to the network
can be determined within a uniform, three-dimensional, global coordinate reference frame. They form a polyhedron which goes around Earth.
As early as 1878 H. Bruns proposed such a concept, later known as the Cage of Bruns. Bruns regarded this objective to be one of the basic problems of scientific geodesy. The idea, however, could not be realized with classical methods, and was forgotten. The geometrical method of satellite geodesy is also called the direct method,
because the particular position of the satellite enters directly into the solution. (2) Satellites can be considered to be a probe or a sensor in the gravity field of Earth. The orbital motion, and the variation of the parameters describing the orbit, are observed in order to draw conclusions about the forces acting. Of particular interest is the relation between the features of the terrestrial gravity field and the resulting deviations of the true satellite orbit from an undisturbed Keplerian motion [3.1.1]. The essential value of the satellite is that it is a moving body within Earth’s gravity field. This view leads to the dynamical method of satellite geodesy.
The main advantage of satellite observations, when compared with classical tech-niques, is that the results refer to the planet Earth as a whole, and that they have a global character by nature. Data gaps play a minor role. Among the first spectacular results were a reasonably accurate value of Earth’s flattening, and the proof that the figure of Earth is non-symmetrical with respect to the equatorial plane (i.e. the pear-shape of Earth, cf. [12.2], Fig. 12.5, p. 517).
In dynamical satellite geodesy orbital arcs of different lengths are considered. When arc lengths between a few minutes and up to several revolutions around Earth are used, we speak of short arc techniques; the term for the use of longer arcs, up to around 30 days and more, is long arc techniques. The orbits are described in suitable geocentric reference frames. The satellite can thus be considered to be the “bearer of
Ne w Station P1 P2 P3 P4 N
Figure 1.3. Orbital method; the satellite is a sensor in Earth’s gravity field
its own coordinates”. The geocentric co-ordinates of the observing ground sta-tions can be derived from the known satellite orbits. This so-called orbital method of coordinate determination is illustrated in Fig. 1.3.
The combined determination of gravity field parameters and geocentric coordinates within the domain of dy-namical satellite geodesy leads to the general problem of parameter determi-nation or parameter estimation. This may include the determination of the ro-tational parameters of Earth (Earth rota-tion, polar motion) as well as other geo-dynamical phenomena (cf. [4.1]). The dynamical method of satellite geodesy is
also characterized as the indirect method, because the required parameters are deter-mined implicitly from the orbital behavior of the satellites.
The distinction geometric–dynamic has, for many years, characterized the develop-ment of satellite geodesy. Today, most of the current techniques have to be considered as combinations of both viewpoints.
A further classification of the observation techniques refers to the relation between the observation platform and the target platform. We distinguish the following groups:
(1) Earth to Space methods
− directions from camera observations, − satellite laser ranging (SLR),
− Doppler positioning (TRANSIT, DORIS), and
− geodetic use of the Global Positioning System (GPS, GLONASS, future GNSS). (2) Space to Earth methods
− radar altimetry, − spaceborne laser, and − satellite gradiometry. (3) Space to Space methods
− satellite-to-satellite tracking (SST).
Earth-bound methods are the most advanced, because the observation process is better under control. With the exception of radar altimetry, the methods mentioned in (2) and (3) are still under development or in their initial operational phase.
1.3
Historical Development of Satellite Geodesy
The proper development of satellite geodesy started with the launch of the first ar-tificial satellite, SPUTNIK-1, on October 4, 1957. The roots of this development can, however, be identified much earlier. If we include the use of the natural Earth satellite, the Moon, then dynamical satellite geodesy has existed since the early 19th century. In 1802, Laplace used lunar nodal motion to determine the flattening of Earth to be f = 1/303. Other solutions came, for example, from Hansen (1864) with f = 1/296, Helmert (1884) with f = 1/297.8, and Hill (1884) with f = 1/297.2 (see Wolf (1985), Torge (2001)).
The geometrical approach in satellite geodesy also has some forerunners in the lu-nar methods. These methods have undergone comprehensive developments since the beginning of the last century. In this context, the Moon is regarded as a geometric target, where the geocentric coordinates are known from orbital theory. The directions be-tween the observer and the Moon are determined from relative measurements of nearby stars, or from occultation of stars by the Moon. Geocentric coordinates are thereby received. Within the framework of the International Geophysical Year 1957/58 a first outcome from a global program was obtained with the Dual Rate Moon Camera, developed by Markovitz (1954). The methods of this so-called Cosmic Geodesy were treated comprehensively in 1960 by Berroth, Hofmann. They also form a considerable part of the classical book of Mueller (1964) “Introduction to Satellite Geodesy”.
Further foundations to satellite geodesy before the year 1957 were given by the work of Väisälä (1946), Brouwer (1959), King-Hele (1958) and O’Keefe (1958). Therefore, it was possible to obtain important geodetic results very soon after the launch of the first rockets and satellites. One of the first outstanding results was the de-termination of Earth’s flattening asf = 1/298.3 from observations of EXPLORER-1 and SPUTNIK-2 by O’Keefe (1958), King-Hele, Merson (1958). Some significant
events during the years following 1957 are
1957 Launch of SPUTNIK-1,
1958 Earth’s Flattening from Satellite Data (f = 1/298.3),
1958 Launch of EXPLORER-1,
1959 Third Zonal Harmonic (Pear Shape of Earth),
1959 Theory of the Motion of Artificial Satellites (Brouwer),
1960 Launch of TRANSIT-1B,
1960 Launch of ECHO-1,
1960 Theory of Satellite Orbits (Kaula),
1962 Launch of ANNA-1B, and
1962 Geodetic Connection between France and Algeria (IGN). By the year 1964, many basic geodetic problems had been successfully tackled, namely the
− determination of a precise numerical value of Earth’s flattening − determination of the general shape of the global geoid
− determination of connections between the most important geodetic datums (to ±50 m).
With hindsight, the development of satellite geodesy can be divided into several phases of about one decade each.
1. 1958 to around 1970. Development of basic methods for satellite observations, and for the computation and analysis of satellite orbits. This phase is characterized by the optical-photographic determination of directions with cameras. The main results were the determination of the leading harmonic coefficients of the geopotential, and the publication of the first Earth models, for instance the Standard Earth models of the Smithsonian Astrophysical Observatory (SAO SE I to SAO SE III), and the Goddard Earth Models (GEM) of the NASA Goddard Space Flight Center. The only purely geometrical and worldwide satellite network was established by observations with BC4 cameras of the satellite PAGEOS.
2. 1970 to around 1980. Phase of the scientific projects. New observation techniques were developed and refined, in particular laser ranging to satellites and to the Moon, as well as satellite altimetry. The TRANSIT system was used for geodetic Doppler positioning. Refined global geoid and coordinate determinations were carried out, and led to improved Earth models (e.g. GEM 10, GRIM). The increased accuracy of the observations made possible the measurement of geodynamical phenomena (Earth ro-tation, polar motion, crustal deformation). Doppler surveying was used worldwide for the installation and maintenance of geodetic control networks (e.g. EDOC, DÖDOC, ADOS).
3. 1980 to around 1990. Phase of the operational use of satellite techniques in geodesy, geodynamics, and surveying. Two aspects in particular are remarkable. Satellite methods were increasingly used by the surveying community, replacing conventional methods. This process started with the first results obtained with the NAVSTAR Global Positioning System (GPS) and resulted in completely new perspectives in surveying
and mapping. The second aspect concerned the increased observation accuracy. One outcome was the nearly complete replacement of the classical astrometric techniques for monitoring polar motion and Earth rotation by satellite methods. Projects for the measurement of crustal deformation gave remarkable results worldwide.
4. 1990 to around 2000. Phase of the international and national permanent services. In particular two large international services have evolved. The International Earth Rotation Service IERS, initiated in 1987 and exclusively based on space techniques, provides highly accurate Earth orientation parameters with high temporal resolution, and maintains and constantly refines two basic reference frames. These are the Inter-national Celestial Reference Frame ICRF, based on interferometric radio observations, and the International Terrestrial Reference Frame ITRF, based on different space tech-niques. The International GPS Service IGS, started in 1994 and evolved to be the main source for precise GPS orbits as well as for coordinates and observations from a global set of more than 300 permanently observing reference stations. At the national level permanent services for GPS reference data have been established and are still growing, e.g. CORS in the USA, CACS in Canada and SAPOS in Germany.
5. 2000 onwards. After more than 40 years of satellite geodesy the development of geodetic space techniques is continuing. We have significant improvements in accuracy as well as in temporal and spatial resolution. New fields of application evolve in science and practice. For the first decade of the new millennium development will focus on several aspects:
− launch of dedicated gravity field probes like CHAMP, GRACE, and GOCE for the determination of a high resolution terrestrial gravity field,
− establishment of a next generation Global Navigation Satellite System GNSS with GPS Block IIR and Block IIF satellites and new components like the Eu-ropean Galileo,
− refinement in Earth observation, e.g. with high resolution radar sensors like interferometric SAR on various platforms,
− further establishment of permanent arrays for disaster prevention and environ-mental monitoring, and
− unification of different geodetic space techniques in mobile integrated geodetic-geodynamic monitoring systems.
1.4
Applications of Satellite Geodesy
The applications of geodetic satellite methods are determined by the achievable accu-racy, the necessary effort and expense of equipment and computation, and finally by the observation time and the ease of equipment handling. A very extensive catalogue of applications can be compiled given the current developments in precise methods with real-time or near real-time capabilities.
Starting with the three basic tasks in satellite geodesy introduced in [1.1], we can give a short summary of possible applications:
Global Geodesy
− general shape of Earth’s figure and gravity field, − dimensions of a mean Earth ellipsoid,
− establishment of a global terrestrial reference frame, − detailed geoid as a reference surface on land and at sea, − connection between different existing geodetic datums, and − connection of national datums with a global geodetic datum. Geodetic Control
− establishment of geodetic control for national networks, − installation of three-dimensional homogeneous networks, − analysis and improvement of existing terrestrial networks,
− establishment of geodetic connections between islands or with the mainland, − densification of existing networks up to short interstation distances.
Geodynamics
− control points for crustal motion,
− permanent arrays for 3D-control in active areas, − polar motion, Earth rotation, and
− solid Earth tides. Applied and Plane Geodesy
− detailed plane surveying (land register, urban and rural surveying, geographic information systems (GIS), town planning, boundary demarcation etc.), − installation of special networks and control for engineering tasks, − terrestrial control points in photogrammetry and remote sensing,
− position and orientation of airborne sensors like photogrammetric cameras, − control and position information at different accuracy levels in forestry,
agricul-ture, archaeology, expedition cartography etc. Navigation and Marine Geodesy
− precise navigation of land-, sea-, and air-vehicles,
− precise positioning for marine mapping, exploration, hydrography, oceanogra-phy, marine geology, and geophysics,
− connection and control of tide gauges (unification of height systems). Related Fields
− position and velocity determination for geophysical observations (gravimetric, magnetic, seismic surveys), also at sea and in the air,
− determination of ice motion in glaciology, Antarctic research, oceanography, − determination of satellite orbits, and
− tomography of the atmosphere (ionosphere, troposphere).
With more satellite systems becoming operational, there is almost no limit to the possi-ble applications. This aspect will be discussed together with the respective techniques. A summarizing discussion of possible applications is given in chapter [12].
1.5
Structure and Objective of the Book
Satellite geodesy belongs equally to fundamental and applied sciences. Both aspects are dealt with; however, the main emphasis of this book is on the observation methods and on the applications.
Geodetic fundamentals are addressed in chapter [2], in order to help readers from neighboring disciplines. In addition, some useful information is provided concerning fundamental astronomy and signal propagation. The motion of near-Earth satellites, including the main perturbations and the basic methods of orbit determination, are discussed in chapter [3], as far as they are required for an understanding of modern observation techniques and applications.
The increasing observational accuracy requires a corresponding higher accuracy in the determination of orbits. In practice, particularly for today’s applications, the user must be capable to assess in each case the required orbital accuracy, and the influence of disturbing effects. This is only possible with a sufficient knowledge of the basic relations in celestial mechanics and perturbation theory. For further studies, fundamental textbooks e.g. Schneider (1981), Taff (1985), or Montenbruck, Gill (2000) are recommended. Special references are given in the relevant sections.
The different observation methods of satellite geodesy are discussed in chapters [4]–[11]. The grouping into currently important observation methods is not without problems, because common aspects have to be taken up in different sections, and be-cause the topical methods develop very quickly. This classification is nevertheless preferred because the user is, in general, working with a particular observation tech-nique, and is looking for all related information. Also a student prefers this type of grouping, because strategies for solving problems can be best studied together with the individual technique. Cross-references are given to avoid unnecessary repetitions. The possible applications are presented together with the particular observation technique, and illustrated with examples. In chapter [12], a problem-orientated sum-mary of applications is given.
The implications of satellite geodesy affect nearly all parts of geodesy and survey-ing. Considering the immense amount of related information, it is often only possible to explain the basic principle, and to give the main guidelines. Recommendations for further reading are given where relevant. For example, an exhaustive treatment of satellite motion (chapter [3]), or of the Global Positioning System GPS (chapter [7]) could fill several volumes of textbooks on their own. As far as possible, references are selected from easily accessible literature in the English language. In addition, some basic references are taken from German and French literature.
2.1
Reference Coordinate Systems
Appropriate, well defined and reproducible reference coordinate systems are essential for the description of satellite motion, the modeling of observables, and the representa-tion and interpretarepresenta-tion of results. The increasing accuracy of many satellite observarepresenta-tion techniques requires a corresponding increase in the accuracy of the reference systems. Reference coordinate systems in satellite geodesy are global and geocentric by nature, because the satellite motion refers to the center of mass of Earth [3]. Terres-trial measurements are by nature local in character and are usually described in local reference coordinate systems. The relationship between all systems in use must be known with sufficient accuracy. Since the relative position and orientation changes with time, the recording and modeling of the observation time also plays an important role.
It should be noted that the results of different observation methods in satellite geodesy refer to particular reference coordinate systems which are related to the indi-vidual methods. These particular systems are not necessarily identical because they may be based on different data and different definitions. Often the relationship be-tween these particular systems is known with an accuracy lower than the accuracy of the individual observation techniques. The establishment of precise transformation formulas between systems is one of the most important tasks in satellite geodesy. 2.1.1 Cartesian Coordinate Systems and Coordinate Transformations
z = z γ xP 0 α x x yP P zP γ γ y y β xP
Figure 2.1. Cartesian coordinate system In a Cartesian coordinate system with the
axesx, y, z the position of a point P is determined by its position vector
xP = xyPP zP , (2.1)
where xP, yP, zP are real numbers (Fig. 2.1).
The transformation to a second Cartesian coordinate system with identi-cal origin and the axesx,y,z, which is generated from the first one by a rotation around thez-axis by the angle γ , can be realized through the matrix operation
with
R3(γ ) =
− sin γ cos γ 0cosγ sinγ 0
0 0 1
. (2.3)
Equivalent rotationsR1around thex-axis and R2around they-axis are
R1(α) = 10 cos0α sin0α 0 − sin α cos α R2(β) = cos0β 0 − sin β1 0 sinβ 0 cosβ . The representation is valid for a right-handed coordinate system. When viewed towards the origin, a counter-clockwise rotation is positive. Any coordinate transformation can be realized through a combination of rotations. The complete transformation is
xP = R1(α)R2(β)R3(γ )xP. (2.4)
The mathematical properties of rotation matrices are described using linear algebra. The following rules are of importance
(1) Rotation does not change the length of a position vector. (2) Matrix multiplication is not commutative
Ri(µ)Rj(ν) = Rj(ν)Ri(µ). (2.5)
(3) Matrix multiplication is associative
Ri(RjRk) = (RiRj)Rk. (2.6)
(4) Rotations around the same axis are additive
Ri(µ)Ri(ν) = Ri(µ + ν). (2.7)
(5) Inverse and transpose are related by
R−1i (µ) = RTi(µ) = Ri(−µ). (2.8)
(6) The following relationship also holds
(RiRj)−1= Rj−1R−1i . (2.9)
The polarity of coordinate axes can be changed with reflectionmatrices
S1= −1 0 00 1 0 0 0 1 ; S2= 10 −1 00 0 0 0 1 ; S3= 10 01 00 0 0 −1 . (2.10)
Finally, the matrix for a general rotation by the anglesα, β, γ is
R=
cosβ cos γ cosβ sin γ − sin β
sinα sin β cos γ − cos α sin γ sinα sin β sin γ + cos α cos γ sinα cos β cosα sin β cos γ + sin α sin γ cosα sin β sin γ − sin α cos γ cos α cos β
. (2.11) The relation between the position vectors in two arbitrarily rotated coordinate systems is then
xP = RxP; xP = RTxP. (2.12)
In satellite geodesy the rotation angles are often very small, thus allowing the use of the linearized form forR. With cos α ∼= 1 and sin α ∼= α (in radians), neglecting higher order terms, it follows that
R(α, β, γ ) = −γ1 γ1 −βα β −α 1 . (2.13)
Although matrix multiplication does not commute (cf. 2.5) the infinitesimal rotation matrix (2.13) does commute.
2.1.2 Reference Coordinate Systems and Frames in Satellite Geodesy In modern terminology it is distinguished between
− reference systems, − reference frames, and
− conventional reference systems and frames.
A reference system is the complete conceptual definition of how a coordinate system is formed. It defines the origin and the orientation of fundamental planes or axes of the system. It also includes the underlying fundamental mathematical and physical models. A conventional reference system is a reference system where all models, numerical constants and algorithms are explicitly specified. A reference frame means the practical realization of a reference system through observations. It consists of a set of identifiable fiducial points on the sky (e.g. stars, quasars) or on Earth’s surface (e.g. fundamental stations). It is described by a catalogue of precise positions and motions (if measurable) at a specific epoch. In satellite geodesy two fundamental systems are required:
− a space-fixed, conventional inertial reference system (CIS) for the description of satellite motion, and
− an Earth-fixed, conventional terrestrial reference system (CTS) for the posi-tions of the observation staposi-tions and for the description of results from satellite geodesy.
2.1.2.1 Conventional Inertial Systems and Frames
Newton’s laws of motion [3.1.2] are only valid in an inertial reference system, i.e. a coordinate system at rest or in a state of uniform rectilinear motion without any acceleration. The theory of motion for artificial satellites is developed with respect to such a system [3].
Space fixed inertial system are usually related to extraterrestrial objects like stars, quasars (extragalactic radio sources), planets, or the Moon. They are therefore also named celestial reference systems (CRS). The definition of a CRS can be based on kinematic or dynamic considerations. A kinematic CRS is defined by stars or quasars with well known positions and, if measurable, proper motions. A dynamical CRS is based on the motion of planets, the Moon, or artificial satellites.
The establishment of conventional celestial reference systems is under the respon-sibility of the International Astronomical Union (IAU). From January 1, 1988, until December 31, 1997, the conventional celestial reference system was based on the ori-entation of the equator and the equinox for the standard epoch J2000.0 (cf. [2.2.2]), determined from observations of planetary motions in agreement with the IAU (1976) system of astronomical constants as well as related algorithms (cf. Seidelmann (ed.) (1992)). The corresponding reference frame was the Fifth Fundamental Catalogue (FK5) (Fricke et al., 1988). ecliptic equator pole Z r M
✗
X α δ S YFigure 2.2. Equatorial system in spherical as-tronomy
The equatorial system at a given epochT0which has been used in spheri-cal astronomy (Fig. 2.2) for many years yields a rather good approximation to a conventional inertial reference system. The origin of the system is supposed to coincide with the geocenterM. The pos-itiveZ-axis is oriented towards the north pole and the positiveX-axis to the First Point of Aries✗. The Y -axis completes a right-handed system. Since Earth’s center of mass undergoes small accel-erations because of the annual motion around the Sun, the term quasi-inertial system is also used.
The traditional materialization of the
above definition for practical purposes is through a catalogue of the positions and proper motions of a given number of fundamental stars. The FK5 is a catalogue of 1535 bright stars, compiled from a large number of meridian observations. The formal uncertainties of the FK5 star positions were about 20 to 30 milliarcseconds at the time of publication (1988). The quality of the FK5 frame is time dependent and is continuously getting worse (de Vegt, 1999; Walter, Sovers, 2000).
Star positions are usually given as spherical coordinates right ascension α and declination δ. The transformation of spherical coordinates α, δ, r into Cartesian
coordinatesX, Y , Z is
X = r cos δ cos α, Y = r cos δ sin α, Z = r sin δ. (2.14)
The reverse formulas are
r =X2+ Y2+ Z2, α = arctan Y
X, δ = arctan Z √
X2+ Y2. (2.15) In spherical astronomyr is usually defined as the unit radius. We may consider the celestial sphere in Fig. 2.2 as the unit sphere and apply the basic formulas of spherical geometry. Detailed information on spherical astronomy can be found in Green (1985) or in textbooks on geodetic astronomy (e.g. Mackie, 1985; Schödlbauer, 2000).
The accuracy of the celestial reference system, realized through the FK5 catalogue, is by far insufficient for modern needs. A considerable improvement, by several orders of magnitude, was achieved with the astrometric satellite mission HIPPARCOS (Ko-valevsky et al., 1997), and with extragalactic radio sources (quasars) via the technique of Very Long Baseline Interferometry (VLBI) which uses radio telescopes [11.1].
In 1991 the IAU decided to establish a new celestial reference system based on a kinematic rather than on a dynamic definition (McCarthy, 2000). The system is called the International Celestial Reference System (ICRS) and officially replaced the FK5 fundamental system on January 1, 1998. The axes of the ICRS are no longer fixed to the orientation of the equator and the vernal equinox, but with respect to distant matter in the universe. The system is realized by a celestial reference frame, defined by the precise coordinates of extragalactic objects (mainly quasars) with no measurable transverse motion. The origin of the ICRS is either the barycenter of the solar system, or the geocenter. The ICRS, hence, consists of the
− Barycentric Celestial Reference System (BCRS), and the − Geocentric Celestial Reference System (GCRS).
The relation between them makes use of general relativity (geodesic precession, Lense-Thirring precession), see McCarthy (2000); Capitaine, et al. (2002).
The International Celestial Reference Frame (ICRF) is a catalogue of the adopted positions of 608 extragalactic radio sources observed via the technique of VLBI. 212 of these objects are defining sources (Fig. 2.3). They establish the orientation of the ICRS axes. The typical position uncertainty for a defining radio source is about 0.5 milliarcseconds (mas). The resulting accuracy for the orientation of the axes is about 0.02 mas (Ma et al., 1997)
In order to maintain continuity in the conventional celestial reference systems the orientations of the ICRS axes are consistent with the equator and equinox at J 2000.0, as represented by the FK5. Since the accuracy of the FK5 is significantly worse than the new realizations of the ICRS, the ICRS can be regarded as a refinement of the FK5 system.
The Hipparcos Catalogue is a realization of the ICRS at optical wavelengths. This catalogue contains 118 218 stars for the epoch J1991.25. The typical uncertainties at catalogue epoch are 1 mas in position and 1 mas / year in proper motion. For the epoch
Figure 2.3. International Celestial Reference Frame ICRF; distribution of the 212 best-observed extragalactic sources (after Ma et al. (1998))
J 2000.0 typical Hipparcos star positions can be estimated in the range of 5 to 10 mas (Kovalevsky et al., 1997; Walter, Sovers, 2000).
With forthcoming astrometric space missions like FAME and GAIA (Walter, Sovers, 2000), see [5.3.3], further improvement of the optical realization of the ICRS to the level of 10 microarcseconds (µas) is expected. Also the link between the ICRF based on radio stars and frames at optical wavelengths will be improved.
For more information on conventional inertial reference systems and frames see e.g. Moritz, Mueller (1987, chap. 9), Seidelmann (ed.) (1992, chap. 2), Walter, Sovers (2000), Schödlbauer (2000, chap. 3), Capitaine, et al. (2002) and [12.4.2].
2.1.2.2 Conventional Terrestrial Systems and Frames
A suitable Earth-fixed reference system must be connected in a well defined way to Earth’s crust. Such a Conventional Terrestrial System (CTS) can be realized through a set of Cartesian coordinates of fundamental stations or markers within a global network.
The origin of an ideal conventional terrestrial reference system should be fixed to the geocenter, including the mass of the oceans and the atmosphere. Thez-axis should coincide with the rotational axis of Earth. Since the geocenter and the rota-tional axis are not directly accessible for observations the ideal system is approximated by conventions. The classical convention for the orientation of axes was based on as-tronomical observations and has been developed and maintained since 1895 by the International Latitude Service (ILS), and since 1962 by the International Polar Mo-tion Service (IPMS) (Moritz, Mueller, 1987). It is established through the convenMo-tional direction to the mean orientation of the polar axis over the period 1900–1905
(Conven-tional Terrestrial Pole (CTP), also named Conven(Conven-tional Interna(Conven-tional Origin (CIO)) and a zero longitude on the equator (Greenwich Mean Observatory (GMO)). GMO is defined through the nominal longitudes of all observatories which contributed to the former international time service BIH (Bureau International de l’Heure).
In 1988 the responsibility for establishing and maintaining both the conventional celestial and terrestrial reference systems and frames, was shifted to the International Earth Rotation Service (IERS), cf. [12.4.2]. Although the IERS results are based on modern space techniques like SLR [8], VLBI [11.1], GPS [7], and Doppler [6], the traditional convention has been maintained within the accuracy range of the classical astronomical techniques in order to provide continuity.
The conventional terrestrial reference system, established and maintained by the IERS, and nearly exclusively used for today’s scientific and practical purposes is the International Terrestrial Reference System (ITRS); its realization is the International Terrestrial Reference Frame (ITRF). The ITRS is defined as follows (Boucher et al., 1990; McCarthy, 2000):
− it is geocentric, the center of mass being defined for the whole Earth, including oceans and atmosphere,
− the length unit is the SI meter; the scale is in context with the relativistic theory of gravitation,
− the orientation of axes is given by the initial BIH orientation at epoch 1984.0, and
− the time evolution of the orientation will create no residual global rotation with regard to Earth’s crust (no-net-rotation condition).
These specifications correspond with the IUGG resolution no. 2 adopted at the 20th IUGG General Assembly of Vienna in 1991. The orientation of axes is also called IERS Reference Pole (IRP) and IERS Reference Meridian (IRM).
The realization of the ITRS, the International Terrestrial Reference Frame (ITRF) is formed through Cartesian coordinates and linear velocities of a global set of sites equipped with various space geodetic observing systems. If geographical coordi-nates (ellipsoidal latitude, longitude, and height) are required instead of Cartesian coordinates(X, Y, Z), use of the GRS80 ellipsoid is recommended (cf. [2.1.4]). The ensemble of coordinates implicitly define the CTP (Z-axis) and the GMO (X-axis).
Nearly every year a new ITRF is realized based on new observations with geodetic space techniques (e.g. Doppler [6], GPS [7], SLR [8], VLBI [11.1]). The result is published under the denomination ITRFxx, where xx means the last digits of the year whose data were used in the formation of the frame. The most recent solution is ITRF2000 (Fig. 2.4), Altamimi et al. (2001). Each particular ITRF is assembled by combining sets of results from independent techniques as analyzed by a number of separate groups. The use of as many different techniques as possible provides a significant decrease of systematic errors.
The establishment of a terrestrial reference frame is not an easy task because Earth’s crust continuously undergoes various deformations. Since today’s geodetic space techniques provide station coordinates at the 1 cm or subcentimeter level, it is necessary to model the various deformations at the mm-level. The main influences are
